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Solvable group

In mathematics, more in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group; the word "solvable" arose from Galois theory and the proof of the general unsolvability of quintic equation. A polynomial equation is solvable by radicals if and only if the corresponding Galois group is solvable. A group G is called solvable if it has a subnormal series whose factor groups are all abelian, that is, if there are subgroups 1 = G0 < G1 < ⋅⋅⋅ < Gk = G such that Gj−1 is normal in Gj, Gj /Gj−1 is an abelian group, for j = 1, 2, …, k. Or equivalently, if its derived series, the descending normal series G ▹ G ▹ G ▹ ⋯, where every subgroup is the commutator subgroup of the previous one reaches the trivial subgroup of G; these two definitions are equivalent, since for every group H and every normal subgroup N of H, the quotient H/N is abelian if and only if N includes H. The least n such that G = 1 is called the derived length of the solvable group G.

For finite groups, an equivalent definition is that a solvable group is a group with a composition series all of whose factors are cyclic groups of prime order. This is equivalent because a finite group has finite composition length, every simple abelian group is cyclic of prime order; the Jordan–Hölder theorem guarantees that if one composition series has this property all composition series will have this property as well. For the Galois group of a polynomial, these cyclic groups correspond to nth roots over some field; the equivalence does not hold for infinite groups: for example, since every nontrivial subgroup of the group Z of integers under addition is isomorphic to Z itself, it has no composition series, but the normal series, with its only factor group isomorphic to Z, proves that it is in fact solvable. All abelian groups are trivially solvable – a subnormal series being given by just the group itself and the trivial group, but non-abelian groups may not be solvable. More all nilpotent groups are solvable.

In particular, finite p-groups are solvable. A small example of a solvable, non-nilpotent group is the symmetric group S3. In fact, as the smallest simple non-abelian group is A5, it follows that every group with order less than 60 is solvable; the group S5 is not solvable — it has a composition series, giving factor groups isomorphic to A5 and C2. Generalizing this argument, coupled with the fact that An is a normal, non-abelian simple subgroup of Sn for n > 4, we see that Sn is not solvable for n > 4. This is a key step in the proof that for every n > 4 there are polynomials of degree n which are not solvable by radicals. This property is used in complexity theory in the proof of Barrington's theorem; the celebrated Feit–Thompson theorem states that every finite group of odd order is solvable. In particular this implies that if a finite group is simple, it is either a prime cyclic or of order. Any finite group whose p-Sylow subgroups are cyclic is a semidirect product of two cyclic groups, in particular solvable.

Such groups are called Z-groups. Numbers of solvable groups with order n are 0, 1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5, 2, 2, 1, 15, 2, 2, 5, 4, 1, 4, 1, 51, 1, 2, 1, 14, 1, 2, 2, 14, 1, 6, 1, 4, 2, 2, 1, 52, 2, 5, 1, 5, 1, 15, 2, 13, 2, 2, 1, 12, 1, 2, 4, 267, 1, 4, 1, 5, 1, 4, 1, 50... Orders of non-solvable groups are 60, 120, 168, 180, 240, 300, 336, 360, 420, 480, 504, 540, 600, 660, 672, 720, 780, 840, 900, 960, 1008, 1020, 1080, 1092, 1140, 1176, 1200, 1260, 1320, 1344, 1380, 1440, 1500... Solvability is closed under a number of operations. If G is solvable, H is a subgroup of G H is solvable. If G is solvable, there is a homomorphism from G onto H H is solvable; the previous properties can be expanded into the following "three for the price of two" property: G is solvable if and only if both N and G/N are solvable. In particular, if G and H are solvable, the direct product G × H is solvable. Solvability is closed under group extension: If H and G/H are solvable so is G.

It is closed under wreath product: If G and H are solvable, X is a G-set the wreath product of G and H with respect to X is solvable. For any positive integer N, the solvable groups of derived length at most N form a subvariety of the variety of groups, as they are closed under the taking of homomorphic images and products; the direct product of a sequence of solvable groups with unbounded derived length is not solvable, so the class of all solvable groups is not a variety. Burnside's theorem states that if G is a finite group of order paqb where p and q are prime numbers, a and b are non-negative integers G is solvable; as a strengthening of solvability, a group G is called supersolvable if it has an invariant normal series whose factors are all cyclic. Si

Molly (1999 film)

Molly is a 1999 romantic comedy-drama film about a 28-year-old woman with autism who comes into the custody of her neurotic executive brother. The film was directed by John Duigan and written by Dick Christie of Small Wonder-fame, stars Elisabeth Shue as the title character, Aaron Eckhart as her older brother, Jill Hennessy. A 28-year-old autistic woman named Molly McKay has lived in an institution from a young age following her parents' death in a car accident; when the institution must close due to budget cuts, Molly is left in the care of her non-autistic older brother, Buck McKay, an advertising executive and perennial bachelor. Molly, who verbalizes little and is obsessed with lining up her shoes in neat rows, throws Buck's life into a tailspin as she runs off her nurses and barges into a meeting at Buck's agency naked. Molly's neurologist, Susan Brookes, suggests an experimental surgery in which genetically modified brain cells are implanted into Molly's brain. While Buck balks at the suggestion, he consents to the surgery and Molly makes a gradual but miraculous "recovery", speaking fluidly and interacting with others in a normal way.

Buck begins taking Molly to social events, like a production of Romeo and Juliet, a baseball game, expensive dinners. However, after a few months, Molly's brain begins to reject the transplanted cells and she begins to regress into her former state. Both Molly and Buck must accept the eventual loss of Molly's "cure" and her regression to her previous state. In the final scene of the film, Buck accepts Molly's autism and vows to remain in Molly's life by creating a room for her at his home that looks just like the room she had at the institution; the film earned US$17,650 during its theatrical run, on a budget of $21 million, making it a box office bomb. Believing the film was unlikely to be a success, the distributors Metro-Goldwyn-Mayer chose to cut their losses and eliminate the film's marketing budget, it was only released on a single weekend in twelve cinemas. Molly received negative reviews from critics. On Rotten Tomatoes, the film holds a 14% "Rotten" approval from film critics, with a rating average of 3.4 out of 10.

The consensus says, "Molly never elevates above uninspired, cliche-ridden moments." At Metacritic, Molly received a weighted mean rating of 21 out of 100 from film critics indicating "generally unfavorable reviews", classified as a unfavorably reviewed film. Charly Molly on IMDb Molly at the TCM Movie Database

Warwick Yates

Warwick Yates is a former Australian rules footballer who played with Geelong in the Victorian Football League. Yates, a ruckman from Lorne, played senior football at Geelong for three seasons, he played 21 league games. In 1974 he began playing for Geelong West in the Victorian Football Association and would go on an play over 200 games for the club, he captained the club in the 1977 season. Yates won his only club best and fairest in 1979. Geelong West made the grand final that year, which Yates missed after getting injured before the preliminary final, it was reported that he broke "a leg while chopping firewood". Appointed captain-coach in 1980, Yates led the club to the Division 1 preliminary finals and had a strong individual season with an equal sixth placing in the J. J. Liston Trophy, he resigned early in the campaign. After Geelong West merged with St Peters, Yates coached the club for its inaugural season in 1989. Warwick Yates's playing statistics from AFL Tables

Shadakshari Settar

Shadakshari Settar was an Indian professor and scholar who had conducted research in the fields of Indian archaeology, art-history, history of religions and philosophy as well as classical literature. Settar studied in Mysuru and Cambridge University. Professor of History and Archaeology, Karnataka University in Dharwad Director, of the National Museum Institute of the History of Art and Museology. President, Indian Council of Historical Research. Dr S Radhakrishnan Chair at the NIAS, India became Professor Emeritus. Honorary Director of the Southern Centre of the Indira Gandhi National Centre for the Arts. Visiting professor at various foreign universities including at Cambridge, Heidelberg, Leiden. Works under Settar's personal authorship comprise four volumes on history of art, two on religion and philosophy, one on human civilization and four on historiography. ಶ್ರವಣಬೆಳಗೊಳ ಸಾವಿಗೆ ಆಹ್ವಾನ ಶಂಗಂ ತಮಿಳಗಂ ಮತ್ತು ಕನ್ನಡ ನಾಡು-ನುಡಿ ಸೋಮನಾಥಪುರ ಬಾದಾಮಿ ಚಾಳುಕ್ಯರ ಶಾಸನ ಸಾಹಿತ್ಯ ಸಾವನ್ನು ಅರಸಿ ಹಳಗನ್ನಡ- ಲಿಪಿ, ಲಿಪಿಕಾರ, ಲಿಪಿ ವ್ಯವಸಾಯ ಹಳಗನ್ನಡ-ಭಾಷೆ, ಭಾಕಾ ವಿಕಾಸ ಮತ್ತು ಭಾಷಾ ಬಾಂಧವ್ಯ ಪ್ರಾಕೃತ ಜಗದ್ವಲಯ Hoysala Sculpture in the National Museum, Copenhagen Sravana Belagola - An illustrated study, Dharwad Inviting Death, Historical Experiment on Sepulchar Hill, Dharwad Inviting Death: Indian Attitude Towards the Ritual Death Pursuing Death: Philosophy and Practice of Voluntary Termination of Life, Dharwad Hampi - A Medieval Metropolis, Bangalore Hoysala temples, Vol I, II, Bangalore Footprints of Artisans in History, Mysore Somanathapura, Bangalore Akssarameru's Kaliyuga Vipartan, Bangalore Archaeological Survey of Mysore: Annual Reports, Vol II-IV, Dharwad Memorial Stones: A Study of the origin and variety, Dharwad-Heidelberg Indian Archaeology in Retrospect, Vol I-IV, New Delhi Construction of Indian Railways, Vol I-III, New Delhi Jalianwala Bagh massacre, New Delhi Pangs of Partition, Vol I-II, New Delhi In 2008 Settar was presented with the Sham.

Ba. Joshi Award for his contributions to historical research. Https:// Google Scholar report

Electoral district of Port Macquarie

Port Macquarie is an electoral district of the Legislative Assembly in the Australian state of New South Wales. It is represented by Leslie Williams of The Nationals, it presently includes parts of coastal Port Macquarie-Hastings City Council and the northeast of the City of Greater Taree. The district includes Lord Howe Island. Port Macquarie was created in 1988, it has been a comfortably safe seat for the National Party. Dating to its time as Oxley, the Port Macquarie area had been held by a conservative party since the return to single-member seats in 1927, had been in National hands for all but six years since 1945; this tradition was broken in 2002, when three-term National member and shadow minister Rob Oakeshott resigned from the party to become an independent. He was handily reelected as an independent in 2003 and 2007. In 2003, he was returned with 82 percent of the two-party vote, making Port Macquarie the safest seat in the legislature. Oakeshott resigned in 2008 to run in a by-election for the federal seat of Lyne, based on Port Macquarie at the time.

He was succeeded by staffer Peter Besseling. However, Besseling was swept out by the Nationals' Leslie Williams at the 2011 state election amid the massive National wave that swept through rural NSW that year; this was due in part to voter anger at Oakeshott's support for the minority federal Labor government. Despite Oakeshott's personal popularity, the Port Macquarie area was still National heartland. "Traditional" two-party matchups between the Nationals and Labor during Oakeshott and Besseling's tenures had always shown Port Macquarie as a comfortably safe National seat. Proving this, Williams retained Port Macquarie in 2015. Despite suffering a 9.8 percent swing against Labor, she still sits on a majority of 19 percent, making Port Macquarie the sixth-safest National seat and the 17th-safest Coalition seat

Gentiana andrewsii

Gentiana andrewsii, the bottle gentian, closed gentian, or closed bottle gentian, is an herbaceous species of flowering plant in the gentian family Gentianaceae. Gentiana andrewsii is native to northeastern North America, from the Dakotas to the East Coast and through eastern Canada, it shares the common name "bottle gentian" with several other species. Gentiana andrewsii blooms in late summer; the flowers are 2 to 4 cm long a rich blue color and bottle shaped with closed mouths. The flowers are clustered in the axis of the top leaves; the stems are 30 to 60 cm long, lax in habit, producing sprawling plants with upturned ends ending with clusters of bee pollinated flowers. The foliage is hairless with a glossy sheen to it; the plant was named in honor of an English botanical artist and engraver. Named infraspecies and hybrids include: Gentiana andrewsii var. andrewsii Gentiana andrewsii var. dakotica Gentiana andrewsii fo. albiflora Gentiana × billingtonii Gentiana × pallidocyanea Closed bottle gentian occurs in wet to dry-mesic prairies and prairie fens in loamy soils, but it can be found in sandy areas, such as near Great Lakes shorelines.

The closed flowers make entrance to nectar difficult for many species of insects. Those strong enough to enter through the top of the flower include the digger bee species Anthophora terminalis and the bumblebee species Bombus fervidus, Bombus griseocollis, Bombus impatiens; the eastern carpenter bee chews a narrow slit at the base of the flower and "steals" nectar without pollinating the plant, a behavior known as nectar robbing. The holes in the petals created by this species allow smaller insects to access the nectar and pollen, including the honeybee, the green sweat bee species Augochlorella aurata and Augochlorella persimilis, the eastern masked bee; this gentian is considered a threatened species in the US states of New York and Maryland